Compensation between Resolved and Unresolved Wave Driving in the Stratosphere:Implications for Downward Control
NAFTALI Y. COHEN, EDWIN P. GERBER, AND OLIVER BUHLER
Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University,
New York, New York
(Manuscript received 17 December 2012, in final form 3 June 2013)
ABSTRACT
Perturbations to the orographic gravity wave parameterization scheme in an idealized general circulation
model reveal a remarkable degree of compensation between the parameterized and the resolved wave driving:
when the orographic gravity wave driving is changed, the resolved wave driving tends to change in the opposite
direction, so there is little impact on the Brewer–Dobson circulation. Building upon earlier observations of such
compensation, an analysis based on quasigeostrophic theory suggests that the compensation between the re-
solved and parameterized waves is inevitable when the stratosphere is driven toward instability by the pa-
rameterized gravity wave driving. This instability, however, is quite likely for perturbations of small meridional
length scale in comparison with the Rossby radius of deformation. The insight from quasigeostrophic theory is
confirmed in a systematic study with an idealized general circulation model and supported by analyses of
comprehensivemodels. The compensation between resolved and unresolvedwaves suggests that the commonly
used linear separation of the Brewer–Dobson circulation into components (i.e., resolved versus parameterized
wave driving) may provide a potentially misleading interpretation of the role of different waves. It may also, in
part, explain why comprehensive models tend to agree more on the total strength of the Brewer–Dobson
circulation than on the flow associated with individual components. This is of particular relevance to diagnosed
changes in the Brewer–Dobson circulation in climate scenario integrations as well.
1. Introduction
The Brewer–Dobson circulation (BDC), first hypoth-
esized by Dobson et al. (1929) and established by Brewer
(1949), is a wave-driven zonal-mean mass transport
circulation in the stratosphere, with air parcels ascend-
ing from the tropical troposphere to the extratropical
stratosphere and descending in the middle and high
latitudes (e.g., Holton et al. 1995). The importance of the
BDC lies in the fact that, together with stratospheric
chemistry, it sets the distribution of stratospheric ozone
and water vapor. Stratospheric ozone is of great impor-
tance as it absorbs the sun’s harmful shortwave radiation
and serves as a greenhouse gas that absorbs longwave
radiation from the ground, and thus is an important factor
in the atmosphere’s radiation budget (e.g., Haynes 2005).
In addition, recent studies show evidence that Southern
Hemisphere tropospheric circulation trends are strongly
influenced by stratospheric ozone concentration changes
(e.g., Thompson and Solomon 2002; Son et al. 2010), and
Solomon et al. (2010) find that stratospheric water vapor
concentration is an important driver of decadal global
surface climate change.
The current physical interpretation of the extratropical
BDC was developed in the 1960s and 1970s [see Haynes
(2005) for an overview] and is usually expressed in terms
of the transformedEulerian-mean (TEM) equations. The
strength of the TEM equations comes from the fact that
its meridional and vertical velocities approximate the
Lagrangian-mean velocities for zonal-mean steady
disturbances (e.g., B€uhler 2009, chapter 11). Moreover,
in the quasigeostrophic (QG) approximation the TEM
equations provide a clear causality of the wave–mean
flow driving, a point that is emphasized in the ‘‘down-
ward control’’ principle (Haynes et al. 1991).
Planetary-scale Rossby waves and small-scale gravity
waves are the primary drivers of the circulation in the
middle atmosphere, where planetary waves dominate in
the stratosphere and gravity waves dominate in the meso-
sphere. Stationary Rossby waves are forced by large-scale
Corresponding author address: Naftali Cohen, Courant Institute
of Mathematical Sciences, 251 Mercer St., New York, NY 10012.
E-mail: [email protected]
3780 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
DOI: 10.1175/JAS-D-12-0346.1
� 2013 American Meteorological Society
mountains and land–sea contrast, and so resolved in at-
mospheric general circulationmodels (AGCMs), whereas
gravity waves that originate from small-scale mountains,
convection, and frontal instabilities cannot be captured in
most of the AGCMs and so need to be parameterized
(Fritts and Alexander 2003; Alexander et al. 2010).
A key aim of this study is to provide a better un-
derstanding of how waves of different scales contribute to
the BDC. The common practice today (e.g., Butchart et al.
2011) is to use the downward control principle to linearly
separate the influence of resolved planetary-scale Rossby
wave driving [the Eliassen–Palm flux divergence (EPFD)]
and parameterized orographic and nonorographic gravity
wave driving (OGW and NOGW, respectively; an extra
‘‘D’’ is added, from time to time, to denote ‘‘driving’’).
Denoting the zonal-mean wave driving by G, then
GTot 5GEPFD 1GOGW 1GNOGW: the total wave driv-
ing is the sum of contributions from the resolved
Rossby wave driving and the unresolved parameterized
orographic and nonorographic gravity wave driving.
The downward control principle relates a unique (up
to a boundary condition) residual-mean meridional
circulation c* to the total wave driving. It is then ar-
gued that the total residual-mean meridional circula-
tion can be viewed as a linear sum of the residual-mean
meridional circulation driven by the different waves.
Hence, cTot* 5cEPFD
* 1cOGW* 1cNOGW
* ; each compo-
nent is associated with the respective wave driving
contribution. Under this assumption, it is found that at
70 hPa, the resolved waves, OGW and NOGW, con-
tribute roughly 70%, 20%, and 10% of the BDC, re-
spectively (Butchart et al. 2011). However, there is
a wide spread between the models in the contributions
from the different types of waves. For example, different
stratosphere resolving models from the Chemistry Cli-
mate Model Validation Activity, phase 2 (CCMVal2),
suggest that the contributions from resolvedwaves,OGW,
and NOGW ranged roughly from 30% to 100%, 0% to
40%, and 0% to 20%, respectively (Butchart et al. 2011).
Climate change studies with chemistry–climate models
consistently predict a strengthening of about 2%decade21
in the BDC throughout the twenty-first century (Butchart
et al. 2006, 2010; Eyring et al. 2010). Overall, the down-
ward control decomposition suggests that at 70hPa, re-
solved wave and parameterized gravity wave driving
account for roughly two-thirds and one-third of the trend,
respectively. There is considerable disagreement between
models, however; in some models the parameterized
OGW even dominate the trend (e.g., Garcia and Randel
2008; Li et al. 2008; McLandress and Shepherd 2009;
Eyring et al. 2010).
This linear decomposition of the forcing gives the im-
pression that each component of the system is responsible
for driving a fraction of the total circulation. Thus, for
example, if models are overestimating changes in gravity
wave driving, should one think they are overestimating
the total trend in the BDC? An earlier study of inter-
actions between orographic gravity wave drag and re-
solved waves by McLandress and McFarlane (1993)
suggests that one should be cautious in drawing such
a conclusion. They found that OGW torque in the me-
sosphere can generate planetary-scale waves, leading to
a nearly equal and opposite resolved wave torque.While
they study in depth the quasi-linear structure of these
interactions, here we concentrate on their zonal-mean
structure, which is relevant to the Brewer–Dobson cir-
culation. We show that there may be substantial in-
teractions between resolved and parameterized waves
throughout the stratosphere: if one component of the
system is altered, the others may act to compensate for
the change. Hence, the decomposition of the BDC into
individual components may not provide an accurate
picture of how each component contributes to the system
as a whole. These strong interactionsmay also explain the
significant spread in the decomposition of the BDC be-
tween the different wave components in the CCMVal2
model climatologies and climate change forecasts.
In section 2, we establish a new modeling framework
to explore the interactions between the resolved waves
and parameterized wave driving in a primitive equation
model of the atmosphere. The model is forced with
a highly simplified forcing that produces a realistic cli-
mate, as established by Held and Suarez (1994) and
Polvani and Kushner (2002), but includes OGW and
NOGW parameterizations from a state-of-the-art
AGCM. The parameterized OGW driving is perturbed
by varying one of its input parameters, as described in
section 3, leading to substantial changes in the parame-
terized OGW driving in the midstratosphere. While an
additive view of the BDC would then suggest a sub-
stantial change in the meridional overturning, the actual
circulation changes very little, as the resolved wave
driving nearly perfectly compensates for the perturba-
tion in the parameterized OGW driving. A number of
similar experiments suggest that the compensation is
fairly robust to many changes in the OGW scheme and
some changes in the NOGW scheme, supporting the
findings of the McLandress and McFarlane (1993) study
based on a quasi-linear QG model.
Analysis of the necessary condition for instability in
the QG framework in section 4 suggests that the com-
pensation is a response of the resolved waves to prevent
instability in the flow driven by the parameterizedOGW
driving. Following the strategy of a ‘‘proof by contra-
diction,’’ we first suppose that the OGWD were not
compensated. We then show that this leads to an
DECEMBER 2013 COHEN ET AL . 3781
unstable mean state. This suggests that some degree
of compensation is required to maintain a ‘‘sensible’’
residual-mean meridional circulation. The key factors
determining the degree of compensation are the am-
plitude and meridional scale of the perturbation: stron-
ger and narrower perturbations are more likely to be
compensated. We then confirm the insight from the QG
theory in a series of systematic experiments with the
fully nonlinear primitive equation GCM.
Even in cases where the parameterized gravity wave
driving is not compensated by the resolved waves (as
explored in section 5), we find that the additive de-
composition of the BDC into its wave-driven compo-
nents can be misleading. The resolved waves can also
respond strongly to changes in the mean state induced
by changes in the gravity wave parameterization. The
strong interactions between the parameterized waves
and resolved waves again suggest that one must be
careful when decomposing the BDC into wave-driven
components. Last, our results summarized and discussed
in the context of comprehensive AGCMs in section 6.
2. A new modeling framework
The AGCM used in this study was developed by the
Geophysical Fluid Dynamics Laboratory (GFDL). It is
exactly the same as that used in Polvani and Kushner
(2002), except for a change in the gravity wave parame-
terization and a minor adjustment to the vertical co-
ordinate. Briefly, the model integrates the dry hydrostatic
primitive equations with pseudospectral numerics. The
model is relaxed toward a simplified perpetual January
radiative equilibrium temperature field to produce re-
alistic tropospheric and stratospheric conditions without
the need of convective or radiative schemes (Held and
Suarez 1994; Polvani andKushner 2002). In particular, the
stratospheric relaxation temperature varies smoothly from
the U.S. standard atmosphere over most of the strato-
sphere to a profile with constant lapse rate over the winter
pole in order to produce a polar night jet. The strength of
the polar night jet is controlled by a single parameter
g with typical values from 4 to 6Kkm21; see Polvani and
Kushner (2002) for details.
We focus on the NorthernHemisphere winter, as this is
the period of maximal coupling between the troposphere
and stratosphere on intraseasonal time scales. In the
winter hemisphere, only large-scale planetary wave
can efficiently propagate into the stratosphere owing to
strong winds in the polar vortex (e.g., Charney and
Drazin 1961). In the dynamical core framework, Gerber
and Polvani (2009) showed that a simple large-scale to-
pography is sufficient to excite an active stratospheric
circulation. We use their configuration with most realistic
coupling—a wavenumber-2 topography with 3-km am-
plitude between 258 and 658N. The strength and structure
of the BDC is controlled directly by the resolved topog-
raphy and indirectly by the strength of the polar vortex.
Gerber (2012) showed that increasing the amplitude of
the resolved topography strengthens the planetary wave
forcing and decreasing the temperature of the polar vor-
tex (parameter g) raises the planetary wave breaking re-
gion, hence deepening the circulation.
Polvani and Kushner (2002) included a crude pa-
rameterization for the mesospheric gravity wave driving
by introducing a Rayleigh friction above 0.5 hPa, acting
on the uppermost layers of the model, which extend to
approximately 0.01 hPa. Drag near the model’s top is
needed to slow down the polar night jet, as the winds in
radiative equilibrium exceed several hundred meters
per second at these altitudes. We have replaced this
crude parameterization with an interactive parameteri-
zation scheme for NOGW (Alexander and Dunkerton
1999) as implemented in GFDL’s atmospheric model,
version 3 (AM3; Donner et al. 2011). We use the same
tuning parameters as in the AM3 settings: the momen-
tum source is represented by a broad spectrum of wave
speeds (half-width of 40ms21) with a resolution of 2ms21
and a single horizontal wavelength of 300km. The ampli-
tude of themomentum source (see appendix) is 0.005Pa in
the Northern Hemisphere and 0.003Pa in the Southern
Hemisphere. The amplitude in the tropics is 0.004Pa in
AM3, but we reduce it by 95% in our dynamical core to
eliminate permanent tropical zonal-mean zonal wind os-
cillations, a hint of which appears in the full atmospheric
model (Donner et al. 2011, their Fig. 14). We note that the
dry model, lacking convection parameterization, is fairly
quiescent in the tropics. Including just part of the wave
spectrum with the gravity wave parameterization leads to
an unrealistic mean state. In addition, we have modified
the NOGW scheme to conserve angular momentum, as
discussed in the appendix.
OGW are parameterized as in GFDL’s AM3
(Pierrehumbert 1987; Stern and Pierrehumbert 1988;
Donner et al. 2011), except that we modify the scheme to
conserve angular momentum, as discussed in the appen-
dix. A key input parameter to the scheme is a measure of
the subgrid-scale mountain height, which quantifies the
amplitude of unresolved topography as discussed in detail
in the following section 3.
All integrations in the study were completed with tri-
angular truncation 42 resolution, corresponding roughly
to a 2.88 grid and 40 hybrid vertical levels. Gerber and
Polvani (2009) found that this resolution was sufficient
to capture stratosphere–troposphere coupling. The
vertical levels are spaced exactly as in Polvani and
Kushner (2002), but we linearly transform from a pure
3782 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
sigma coordinate at the surface to pure pressure co-
ordinate above 200 hPa. Unless otherwise specified,
experiments are integrated for 10 000 days, excluding
800 spinup days.
To gain confidence in the new model settings, we
compare a few of the main stratospheric features of the
old model configuration with the Rayleigh drag and g 54Kkm21 (Gerber and Polvani 2009) to the new model
configuration with just the NOGW parameterization
and g 5 6Kkm21. We have increased g in order to get
the same polar night jet structure, as the NOGWscheme
produces more wave drag in the midstratosphere. The
main characteristics of the zonal-mean zonal wind
and residual-mean mass streamfunction circulation
are similar. In particular, the forcing of the NOGW in
the mesosphere (Fig. 1d) is similar to the forcing of
the upper-layer Rayleigh drag (Fig. 1c). Note that in the
new climate configuration the residual-mean mass cir-
culation has deepened in the stratosphere (Figs. 1e,f).
This was primarily caused by the increase in g, which
controls the strength of the polar vortex, as discussed in
Gerber (2012).
Stratosphere–troposphere coupling, as measured by
the frequency of stratospheric sudden warming (SSW)
events (both minor and major), exhibits similar vari-
ability in both configurations—on average one major
warming event every 200–300 days, consistent with the
observed frequency of SSWs (Charlton and Polvani
2007), although all warmings are split vortex events,
given the wavenumber-2 topographic forcing. The
breakdown of the polar vortex at 10 hPa precedes
a persistent shift of the tropospheric northern annular
mode (NAM) toward a low-index state in the new
model, as in observations (Baldwin and Dunkerton
2001). Last, the annular mode time scale of variability in
the new model captures the observed increase in per-
sistence in the lower stratosphere, as with the old con-
figuration (Gerber and Polvani 2009). In summary, the
new configuration preserves the key quantities of the
stratosphere–troposphere system, but with an improved
representation of gravity wave driving. The new model
setting with its physically consistent NOGW parame-
terization and surface topography defines the default
configuration of the study.
3. Compensation between resolved andparameterized wave driving
To study the interactions between resolved planetary
waves and parameterized gravity waves, we inten-
tionally perturb the OGW driving of the stratosphere.
McLandress and McFarlane (1993) induced pertur-
bations to the OGW driving in the mesosphere by
suppressing interaction between their parameterization
and the resolved flow and shifting the resolved planetary
wave source. Focusing rather on the resolved flow,
Gerber (2012) showed that the level of resolved wave
breaking is influenced by the thermal forcing of the
polar vortex, which sets zonal wind structure and thus
the critical layer for stationary waves. In some analogy
to both approaches, we perturb the OGW driving by
shifting the location of the critical layer for stationary
gravity waves through modification of an input param-
eter in the OGW scheme: the subgrid-scale mountain
height, which represents the amplitude of unresolved
topography.
The spatial structure of OGW source was chosen to
have a global wavenumber-2 pattern with a peak am-
plitude of 240m, as shown in Fig. 2. This broadscale
pattern was chosen to maximize interactions with the
resolved planetary waves, which were also forced at
wavenumber 2. The amplitude was chosen to preserve
the global average of the subgrid-scale topography
height, as set in integrations of AM3 at equivalent hor-
izontal resolution. With these choices, the phase angle
between the resolved and subgrid-scale pattern is the
sole remaining free parameter in the model. Here we
focus on the extreme configurations, with the two pat-
terns positively or negatively correlated. Additional
experiments with intermediate configurations (not
shown) revealed that these extremes captured the full
range of interactions. In one extreme, the ‘‘positive
correlation’’ configuration, we align the subgrid-scale
mountain height with the ridges of the resolved topog-
raphy (08 phase shift; Fig. 2a), while in the other, the
‘‘negative correlation’’ configuration, we shift the subgrid-
scale mountain height to the valleys (1808 phase shift;
Fig. 2b).
Longitude–height cross sections of the zonal-mean
zonal wind at the maximum amplitude of the resolved
mountain (458N) in Figs. 2c and 2d show how changing
the phase between the resolved and unresolved topog-
raphy modifies the OGWD. In the negative-correlation
integration a critical layer for stationary waves (where
u5 0) is located in the stratosphere over the valleys, the
OGW source region (Fig. 2d), while the wind remains
positive at all levels over the ridges, the OGW source
region in the positive-correlation integration (Fig. 2c).
The effect of critical layers in the Pierrehumbert
(1987) scheme is parameterized by limiting the flux by
the square of the zonal velocity. It follows that the
parameterized momentum flux generated in the negative-
correlation integration dissipates lower in the stratosphere
compared to the flux in the positive-correlation in-
tegration. Figure 3 shows the total impact of the phase
shift on the time- and zonal-mean OGW driving. In the
DECEMBER 2013 COHEN ET AL . 3783
negative-correlation integration (Fig. 3b), the parame-
terized wave driving is trapped in the lower strato-
sphere, and the wave driving above 70 hPa is extremely
weak. In the positive-correlation integration (Fig. 3a),
however, there is a substantial drag on the upper
stratosphere.
The difference between the OGWD in the positive
and negative integrations is shown in Fig. 4a. The am-
plitude of the zonally integrated perturbation is quite
significant, on the order of 109N, and can be put into
context by considering the residual-mean mass circula-
tion implied by downward control, as shown in Fig. 4b.
FIG. 1. A comparison between model configurations with Rayleigh friction, the ‘‘old’’ model used in Gerber and
Polvani (2009), and the Alexander and Dunkerton (1999) NOGW parameterization, the ‘‘new’’ model used in this
study: (a),(b) the time- and zonal-mean zonal winds (m s21), (c),(d) the time- and zonal-mean parameterized gravity
wave driving (109N), and (e),(f) the residual-mean mass streamfunction (109 kg s21).
3784 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
The total mass transport associated with this perturba-
tion, assuming no change in the resolved wave driving,
would be on the order of 109 kg s21, a substantial fraction
of the total transport by the BDC, which is approximately
1.53 109 kg s21 at 508N and 70hPa (as shown in Fig. 1e).
The actual change in the residual circulation, however, is
extremely weak, as illustrated in Fig. 4c. The residual
circulation appears to ignore the torque entirely.
This does not, however, imply a problem with down-
ward control, as one must also consider the response
of the resolved wave driving. Figure 5a shows the dif-
ference in the resolved wave driving between the pos-
itive and negative integrations (i.e., the difference
between Figs. 3c and 3d). It largely cancels out theOGW
perturbation (Fig. 4a), so that the total change in wave
forcing, DGEPFD 1DGOGW 1DGNOGW, is very small.
Note how this compares particularly well with the com-
pensation observed in themesophere byMcLandress and
McFarlane (1993, their Fig. 10), where they compared
integrations in which the interaction between the pa-
rameterized wave drag and the resolved flow was tog-
gled on and off. Fourier decomposition of the EPFD in
both integrations reveals that the response of the re-
solved waves to the OGWperturbation, as quantified by
the difference in the EPFD, is dominated by planetary
wavenumber 2 (Fig. 5b). Overall, the result is that the
net forcing on the stratosphere is almost the same in
the positive- and negative-correlation integrations, and
FIG. 2. The impact of the phase shift between the resolved and unresolved topography in the positive- and negative-
correlation integrations. (a),(b) The structure of the resolved and unresolved topography. The black contours show
the resolved, large-scale wavenumber-2 topography, with a maximum amplitude of 3 km; solid contours denote
ridges and dashed contours denote valleys. The red shading shows the unresolved, wavenumber-2, subgrid-scale
mountain height, which is an input parameter for the OGW scheme; the parameter is nonnegative, varying from
0 (white) to 240m (darkest red shading), and quantifies the amplitude of unresolved topography within each grid box.
(a) The positive-correlation configuration, where the subgrid-scale mountain height is largest over the ridges, and (b)
the negative-correlation configuration, where the subgrid-scale mountain height is largest over the valleys. (c),(d)
The time-mean zonal wind (m s21) as a function of longitude and height at 458N. The thick black contours at the
bottom of the figures denote the large-scale resolved topography and the black thin contours denote the OGW
momentum flux, with contours varying from 23.5 3 1023 to 23.5 3 1027 Pa.
DECEMBER 2013 COHEN ET AL . 3785
consequently the meridional overturning circulation
does not change as shown in Fig. 4c.
As highlighted in Figs. 3e and 3f, however, the change
in OGW driving does have a significant impact on the
zonal-mean zonal wind. The difference in the zonal winds
is shown in Fig. 4d, revealing a dipole structure of am-
plitude 20ms21 centered about theOGWDperturbation.
Close inspection of theOGWDandEPFDfields (Figs. 4a
and 5a) reveals that while compensation is nearly
exact at the center of the OGW perturbation, there
are slight differences on the flanks. As also observed
by McLandress and McFarlane (1993), the zonal wind
is more sensitive to changes in the momentum budget
than the meridional overturning circulation.
FIG. 3. (top) The time- and zonal-mean OGW driving GOGW in the (a) positive-correlation and (b) negative-
correlation integrations (109N); (middle) the time- and zonal-mean EPFDGEPFD in the (c) positive-correlation and
(d) negative-correlation integrations (109N); and (bottom) the time- and zonal-mean zonal wind u in the (e) positive-
correlation and (f) negative-correlation integrations (m s21).
3786 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
Figure 6 shows the residual-mean streamfunction at
70 hPa for the negative and positive integrations. While
these integrations essentially have the same meridional
circulation, they tell a very different story on the relative
importance of the resolved and parameterized waves
to the stratospheric mass transport. In the positive-
correlation integration (Fig. 6a), orographic gravity waves
drive over half of the stratospheric mass flux across 508N,
while in the negative-correlation integration (Fig. 6b),
they play an insignificant role. Viewed together, however,
these plots suggest that either interpretation is likely to be
misleading. The total circulation is about the same in both
integrations, despite large changes in the OGW scheme,
suggesting that large-scale constraints govern the total
meridional overturning circulation.
Robustness of the compensation response
We define a heuristic measure for the compensation
that can be applied for different perturbation in-
tegrations. In general, given a perturbationP to a system
in equilibrium, the system may react with a response R
in order to equilibrate again. In the following, we define
a measure that aims to quantify the relation between the
perturbation and the response. In our system, P(xi) is
a perturbation to the zonal-meanwave driving, where xi is
a generic spatial coordinate. The responseR(xi) is defined
as the net change in the other components of the zonal
wave driving. For example, in the positive- and negative-
correlation integrations shown in Figs. 2–6, P5DGOGW,
the change in the OGWD, and R5DGEPFD 1DGNOGW,
the change in the EPFD and the NOGWD. The degree
of compensation C between the P and R is defined as
the scaled covariance between the perturbation and
the response,
C5 12�i[P(xi)1R(xi)]
2
�iP2(xi)1 �
iR2(xi)
52
2�i[P(xi)R(xi)]
�iP2(xi)1 �
iR2(xi)
,
(1)
FIG. 4. (a) The difference in the time- and zonal-mean OGW driving DGOGW between the positive- and negative-
correlation integrations (109N). (b) The difference in the residual circulationDcOGW* (109 kg s21) associated with the
difference in the OGWdriving, as computed by downward control. (c) The difference in the total residual circulation
DcTot* (109 kg s21) and (d) the difference in the zonal-mean zonal wind Du (m s21).
DECEMBER 2013 COHEN ET AL . 3787
where the summation is constraint to the region in
which jP(xi)j. 0:1 maxi
jP(xi)j. The threshold of 0.1 is
established so thatC ismeasured only in the regionwhere
jPj is greater than 10% of its maximum absolute value.
This was done because in regions where the perturbation
is weak, noise in the response can obscure the signal. We
also restrict our analysis to regions above 70 hPa, as
changes in the upper troposphere–lower stratosphere
can overwhelm the rest of the stratosphere.
With this definition, if the response is equal and op-
posite to the perturbation (i.e., R 5 2P), we have per-
fect compensation andC5 1. IfR5 0, or more generally
is uncorrelated with the perturbation, then C5 0 (there
is no compensation) and if R 5 P, then C 5 21; the
system amplifies the perturbation. In the case of the
positive- and negative-correlation integrations we ob-
serve a high degree of compensation: C 5 0.95 6 0.01.
This case is labeled as ‘‘control run’’ in Fig. 7. We em-
phasize that this metric best reflects changes in the
meridional overturning circulation; even at 0.95, dif-
ferences in the total wave forcing in these interactions
does lead to nontrivial changes in the zonal wind, as
seen in Figs. 3e and 3f.
The uncertainty of the compensation was computed
using the bootstrap and the moving-blocks bootstrap
methods (Efron and Tibshirani 1994; Wilks 1997). We
used the bootstrap with 200-member resampling and the
moving-blocks method with 200 resamples and a block
size of 100 days in order to retain the time correlation in
the data. The different methods yield almost identical
results; thus we will show only the former. Although it
turns out that the need to bootstrap the data to de-
termine the uncertainty is insignificant when the com-
pensation is high, it is important for cases of weak
compensation. For example, when one tries to assess the
effect of a weak wave forcing, the natural variability can
more easily overwhelm the forcing effect.
We first verify that compensation is not an artifact of
resolution. Integrations with double the vertical reso-
lution (80 levels) and with double horizontal resolution
(T85 spectral truncation) yield a virtually identical de-
gree of compensation between the positive-and negative-
correlation integrations.
We next verify that the compensation is not an artifact
of the particular wavenumber-2 topography of our default
configuration integrations. In particular,McLandress and
McFarlane (1993) focused on the mesosphere, where
zonal asymmetries are dominated by wavenumber 1.We
run similar simulations, but with lower boundary set-
tings of wavenumber 1 (k 5 1) and wavenumber 3 (k 53), keeping the same amplitude of 3 km. [Similar con-
figurations were explored inGerber and Polvani (2009).]
The subgrid-scale mountain height parameter is given
a large-scale pattern with the same wavenumber k as
the resolved topography, again with a maximum height
of 240m. We consider cases with positive correlation
(where the resolved and unresolved topographies align
with each other) and negative correlation (ridges align
with the valleys). As before, P and R are the difference
in GOGW and GEPFD 1GNOGW between the positive-
correlation and negative-correlation integrations (see
Table 1). Figure 7 (see labels ‘‘k 5 1’’ and ‘‘k 5 3’’)
show that there is a consistently high degree of com-
pensation in these experiments. We can conclude that
the compensation is a fairly generic feature in the
model.
Holton (1984) suggested that the spatial variation of
the OGWD can lead to the generation of planetary
FIG. 5. (a) The difference in the resolved wave forcing DGEPFD
between the positive- and negative-correlation integrations in the
time and zonal mean (109N). (b) The contributions of planetary
wavenumber 2 to the difference in the resolved wave forcing
(109N).
3788 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
waves in the mesosphere. Even though the wave driving
is determined separately for each column, the structure
of the subgrid-scale mountain height parameter gives
the resulting drag significant zonal structure relative to
the climatological stationary planetary waves as seen—
for example, in our integrations in Figs. 2c and 2d. The
parameterized wave forcing can thus be a source of
planetary waves, providing a resolved Eliassen–Palm flux
divergence that could compensate the parameterized
torque. McLandress andMcFarlane (1993) found further
evidence for the generation of planetary waves by the
OGWD, although their analysis suggested a different
mechanism than initially proposed by Holton (1984).
To assess the significance of the spatial and temporal
structure of the parameterized wave driving, we turned
off the OGW parameterization and ran simulations us-
ing a steady specified torque derived from the OGW
forcing of the positive-correlation (with k 5 2) simula-
tion. Specifically, we ran three simulations with specified
forcing by applying (i) the time mean and (ii) the time
and zonal mean, and by (iii) using the time-mean forcing
but shifting its phase by 1808. Note that the latter ex-
periment is similar, in structure, to the sensitivity case D
in McLandress and McFarlane (1993), except that they
instead shift the resolved waves. Figure 7 (see labels
‘‘time-mean forcing,’’ ‘‘time- and zonal-mean forcing,’’
FIG. 6. The residual-mean streamfunction at 70 hPa, as a function of latitude, for the (a) positive-correlation and
(b) negative-correlation integrations (109 kg s21). ‘‘Direct’’ refers to the total residual-mean streamfunction com-
puted directly from the definition of the residual-mean velocities. ‘‘EPFD1GWD’’ refers to the total residual-mean
mass streamfunction computed by downward control, while ‘‘EPFD,’’ ‘‘NOGWD,’’ and ‘‘OGWD’’ refer to the
residual-mean streamfunctions associated with each of the wave components: resolved, nonorographic, and oro-
graphic gravity wave driving (the thick black curve is thus the sum of the blue, green, and red curves).
FIG. 7. (a) The compensation metric C for the various integrations. Each bar corresponds to a difference between two
integrations as listed inTable 1. For example, the control run is the difference between integrations 3 and 4 in Table 1. The
error bars correspond to one standard deviation in C. (b) The bootstrap kernel density estimate for the integrations
labeled by ‘‘L 5 2.5,’’ ‘‘L 5 5.0,’’ . . . , ‘‘L 5 25.0.’’ The colored bars in (a) and the colored curves in (b) match.
DECEMBER 2013 COHEN ET AL . 3789
and ‘‘shifted time-mean forcing’’) shows, however, that
each of these experiments exhibits the same high degree
of compensation: the time-mean forcing experiment
suggests that variability of the OGW in time is not im-
portant, and the zonal-mean and shifted time-mean ex-
periments suggest that the zonal structure is also not
important. The case in which the OGW forcing is ap-
plied as a time- and zonal-mean torque, in particular,
rules out the possibility that compensation depends on
the generation of planetary waves by the parameterized
torque.
While the structure of the OGWD does not appear to
matter, a second question is how the structure of the
resolved waves and background flow affect the com-
pensation. We address this question by altering the
planetary wave source and by modifying the mean state
of the stratosphere, hence changing the propagation
properties of the planetary waves.
In the first experiment, we alter the topographic
planetary wave source by removing the large-scale to-
pography, consequently reducing the stratospheric
planetary wave activity and decreasing planetary waves
breaking in the stratosphere (Gerber 2012). This is
similar to the sensitivity case C in McLandress and
McFarlane (1993), where they suppressed the resolved
wave forcing. In the second experiment we modify the
mean state of the stratosphere. Using the parameter g,
we completely remove the polar night jet, so that the
planetary waves dissipate or reflect before they can en-
ter the stratosphere (Gerber and Polvani 2009). To
prevent the OGW from changing, we specify a steady
torque equal to the time-mean OGW driving from the
positive-correlation integration (with k 5 2), and com-
pare this integration to an integration with the time-
mean OGW driving from the negative-correlation
integration (with k5 2) (see Table 1). Figure 7 (see labels
TABLE 1. Summary of the configurations for the different integrations, where an overbar denotes a zonal mean and square brackets
denote a time mean. In the second-to-left column, the wavenumber and phase shift, relative to the resolved topography, of the pattern
governing the subgrid-scale topography height (SSTH) is specified. In the right column, [int. 3] refers to the time-meanOGW torque from
integration 3, and so forth. Multiplication by 2, ½, 1/4, and 1/8 denotes that the forcing above 70 hPa was multiplied by 2, ½, 1/4, and 1/8.
Integration
number Integration name g
Wavenumber,
resolved
topography
Rayleigh
drag NOGW OGW
SSTH
wavenumber
(phase shift)
Prescribed
torque
1 Default ‘‘new’’ model 6 2 U
2 Default ‘‘old’’ model 4 2 U
3 Positive correlation with k 5 2 6 2 U U 2 (08)4 Negative correlation with k 5 2 6 2 U U 2 (1808)5 Positive correlation with k 5 1 6 1 U U 1 (08)6 Negative correlation with k 5 1 6 1 U U 1 (1808)7 Positive correlation with k 5 3 6 3 U U 3 (08)8 Negative correlation with k 5 3 6 3 U U 3 (1808)9 Time mean, positive correlation
with k 5 2
6 2 U [int. 3]
10 Time and zonal mean, positive
correlation with k 5 2
6 2 U [int.3]
11 Shifted time mean, positive
correlation with k 5 2
6 2 U [int. 3] (908)*
12 Flat, positive correlation 6 Flat U [int. 3]
13 Flat, negative correlation 6 Flat U [int. 4]
14 No polar night jet, positive
correlation
— 2 U [int. 3]
15 No polar night jet, negative
correlation
— 2 U [int. 4]
16 23 stratospheric forcing,
positive correlation
6 2 U 2 [int. 3]
17 0.53 stratospheric forcing,
positive correlation
6 2 U (1/2) [int. 3]
18 0.253 stratospheric forcing,
positive correlation
6 2 U (1/4) [int. 3]
19 0.1253 stratospheric forcing,
positive correlation
6 2 U (1/8) [int. 3]
20 ‘‘New’’ model with g 5 4 4 2 U
21 ‘‘Old’’ model with g 5 6 6 2 U
* In this integration, the time-mean torque from integration 3 was shifted by 908.
3790 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
‘‘flat resolved topography’’ and ‘‘no polar night jet’’)
show almost perfect compensation for these two experi-
ments. It follows that the climatological structure of the
resolved waves plays also only a limited role in the
compensation process.
Last, we test how the amplitude of the parameterized
forcing affects the compensation. We increase/decrease
the stratospheric prescribed time-mean forcing and ob-
serve the change in compensation. We run four simu-
lations in which we use again the time-mean OGW
driving from the positive-correlation integration with
k5 2, but with double, half, one-quarter, and one-eighth
the time-mean stratospheric forcing above 70 hPa. Per-
turbation will be defined as before, by comparing to an
integration with the time-mean forcing from the negative-
correlation integration (see Table 1). Figure 7 (see labels
‘‘23,’’ ‘‘1/23,’’ ‘‘1/43,’’ and ‘‘1/83 stratospheric forcing’’)
shows that the compensation decreases as the amplitude
of the OGW perturbation decreases.
To summarize, our integrations show that resolved
Rossby waves act to compensate perturbation of the
parameterized OGW, so that the residual-mean mass
circulation does not change. We find that the process is
fairly robust to significant changes in both theOGWand
the resolved wave driving, suggesting that the mecha-
nism does not depend on the zonally asymmetric struc-
ture of the flow or the OGW. It is only weakened when
we reduce the amplitude of the perturbation.At first, the
last group of results puzzled us, as it appears that there is
less compensation in instances when the resolved waves
have less of a perturbation to compensate. However, the
fact that compensation becomes weaker for weaker
perturbations led us to suspect that the OGW forcing
may be generating instability. A hint can be seen in Fig.
3c, where a positive EPFD is located at the region of
compensation, suggesting a source of wave activity in
the stratosphere. Thus, we hypothesize that compensa-
tion occurs when the parameterized forcing destabilizes
the stratosphere and that the planetary waves adjust to it
in order to stabilize the flow again. In the next section,
we examine this hypothesis by studying the necessary
condition for instability as a function of the wave driving
using the QG TEM equations.
4. Constraints on the wave driving in the QG TEMequations
We will proceed largely by developing a proof by
contradiction. We suppose that there is no compensa-
tion between the resolved and unresolved waves and
show that the resulting zonal-mean flow is unstable.
Thus, the resolved meridional circulation must respond
to maintain stability. We find that it does so, producing
resolved wave forcing that is roughly equal and opposite
to the parameterized wave forcing.
a. Theory
A necessary condition for instability in the QG
framework is that the zonal-mean meridional QG po-
tential vorticity (QG-PV) gradient changes sign some-
where in the domain. Denoting the zonal-mean PV by q,
in Cartesian pressure coordinates (x, y, p) the QG-PV
gradient is
qy5b2 uyy2 (�up)p . (2)
Terms on the right-hand side correspond to the merid-
ional gradient in the planetary vorticity, relative vor-
ticity, and vorticity stretching, respectively (e.g., Edmon
et al. 1980; Vallis 2006). An overbar represents a zonal
mean and a subscript a partial derivative,b5 2V cosf0/a,
whereV is Earth’s angular velocity,f is the latitude,f0 is
a reference latitude, a is Earth’s radius, u is the zonal
wind, and � is a related to the stratification and discussed
in further detail below. We follow the downward control
argument (Haynes et al. 1991) to compute the zonal-
mean flow (winds and temperature) in response to
a steadywave driving. The goal is to express the necessary
condition for instability in terms of the wave driving
alone. In this way we can determine whether a steady
stable solution is possible for a given parameterized wave
driving.
The steady QG-TEM equations with simple New-
tonian relaxation of the temperature are (e.g., Andrews
and McIntyre 1978; Edmon et al. 1980)
2f0y*5G , (3a)
yy*1vp*5 0, (3b)
v*up 1u2 urtr
5 0, and (3c)
f0up 2pk21R
pk0uy5 0, (3d)
where f0 5 2V sinf0 is the Coriolis acceleration, y* and
v* are the residual-mean meridional and vertical winds,
G is the zonal-mean wave driving, and u is the zonal-
mean potential temperature, assuming up 5 up(p)
(Andrews et al. 1983). Also, ur is the radiative equilib-
rium potential temperature that is in thermal wind bal-
ance with zonally uniform wind field ur, tr is the
relaxation time scale, and � in (2) is �52f 20 pu/RT up;
� is a function of p alone, is strictly positive, and cap-
tures changes in the stratification. Finally, p0 5 103 hPa,
DECEMBER 2013 COHEN ET AL . 3791
k 5 R/cp ’ 2/7, where cp is the specific heat at constant
pressure, and R is the dry gas constant.
Following the downward control derivation, using
(3a) in (3b) we get
y*52G
f00
v*(y,p)51
f0
ðpp1
Gy(y, s) ds1v*(y,p1) . (4)
Taking the y derivative of (3c) and using (4) yields
(ur 2 u)y5 trvy*up
5trup
f0
ðpp1
Gyy(y, s) ds1 trupvy* (y, p1) . (5)
Multiplying (5) by f0/up and taking the p derivative gives
trGyy5 f0
"(ur 2 u)y
up
#p
5 [�(u2 ur)p]p , (6)
where the equality on the right follows using (3d).
Equation (6) provides the essence of the downward
control: the mean flow fields u and u are solely de-
termined by thewave forcingG (up to a boundary term).
In a stable dynamically equilibrated stratosphere, the
background qy is positive (e.g., in regions with no shear
flow, qy 5b. 0) and so we write the necessary condi-
tion for instability as qy , 0. Let us consider a mean
wave driving G0 that determines the zonal-mean fields
u0 and u0 such that qy 5 q0y $ 0, where q0 denotes the
zonal-mean QG-PV of the mean state. For example,
Fig. 8 shows q0y over the Northern Hemispheric strato-
sphere for the default integration with the NOGW (sim-
ulation 1 in Table 1); q0y is strictly positive and its overall
amplitude is on the order of b. Now, consider a pertur-
bation to the stable system. We denote the perturbed
wave driving as G1 and the resultant flow as u1 and u1.
Under the assumption that the wave forcing is linearly
additive, with no interactions between the wave forcings,
G5G0 1G1, the necessary condition for instability in (2)
becomes
qy5 q0y1 q1y, 00q0y, u1yy1 (�u1p)p , (7)
where q1y is defined to be
q1y [ 2 u1yy 2 (�u1p)p . (8)
In words, the necessary condition for instability is that
the perturbed QG-PV meridional gradient overwhelms
the existing meridional gradient. Equation (6) can be sep-
arated for a basic-state balance governed byG0 and a bal-
ance due to the perturbationG1, where the latter balance is
trG1yy5 (�u1p)p . (9)
Using standard dimensional analysis technics (e.g.,
Barenblatt 1996), let the perturbed wave driving G1
scale with amplitude A, let L and H be the meridional
and vertical scales on which the wave driving varies,
respectively, and let u1 scale with U. The scale of the
background QG-PV gradient q0y is denoted as Qy. To
make analytical progress, we simplify � by assuming con-
stant stratification N2 52g2rup/u and that the tempera-
ture is equal to the reference temperature T5T0. Thus,
� simplifies to ( f0gp/RT0N)2 5 (p/Ld)2, whereLd5NHr/
f0 is the Rossby radius of deformation and Hr 5 RT0/g is
the density height scale. It follows from (9) thatu1 scales as
u1 }U } trAL2d
L2
H2
H2r
. (10)
The same scaling analysis can be found in Garcia (1987)
and Haynes et al. (1991), though in the latter they denote
Ld/L asHr/HR, whereHR5 f0L/N is the ‘‘Rossby height’’
for the problem. Using (9) and (10), we scale each term in
the necessary condition for instability [(7b)] to get
Qy ,trA
L2max
(L2d
L2
H2
H2r
, 1
). (11)
Note that the factorL2dH
2/L2H2r is the ratio between the
amplitude of the QG-PV associated with meridional
FIG. 8. The time and zonal mean of the meridional gradient of
theQG-PV q0y (m21 s21) in the default integrationwith theNOGW
and g 5 6Kkm21. The black contours denote the location and
strength of the wave forcing G1 discussed in section 4.
3792 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
gradient in the relative vorticity 2u1yy and vortex
stretching 2(�u1p)p.
d For L � Ld, when L is large compared to the Rossby
radius of deformation, the vertical gradient in u1,
associated with the vortex stretching, dominates the
PV gradient and (11) yields a simple condition on the
necessary condition for instability:
A.QyL
2
tr. (12)
In this limit the amplitude must be fairly large to
satisfy the necessary condition for instability.d For L � Ld, when L is small compared to Ld, the
relative vorticity gradient dominates and (11) yields
the following necessary condition for instability:
A.QyL
4
trL2d
H2r
H2. (13)
Here, the critical amplitude for instability decreases
rapidly with the meridional scale of the perturbation,
and so is extremely sensitive to L. In addition, the
critical amplitude is inversely proportional to the square
of the wave forcing height scale: the larger the vertical
extent, the smaller the critical amplitude for instability.d If L 5 Ld and H 5 Hr, the condition is
A[Ac5QyL
2d
tr. (14)
If we suppose that Qy scales as b, then we get a rough
estimate of the critical amplitudeAc’ 83 1026m s22’1ms21 day 21, based on the parameters consistent with
our model,N5ffiffiffi5
p3 1022 s21,H5 7km, tr5 40 days,
and where the torque is centered around f 5 508N.
WhenA and L are such that q0y 1q1y , 0, as in (7a), the
only way that a stable equilibrated limit can be achieved
is if a resolved wave driving is generated such that
q0y1 (q1y1 qRy). 0, (15)
where qR stands for the contribution of a perturbed re-
solved Rossby wave to the QG-PV meridional gradient.
In an OGW parameterization, L is set primarily by the
spatial extent of the subgrid-scale mountain height and
surface winds. In practice, both tend to vary on a much
shorter scale than the deformation radius Ld of the
stratosphere, which is about 50% larger than in the tro-
posphere because of increased stratification. Typical am-
plitudes for the OGWDare on the order of 1ms21 day 21
(Alexander et al. 2010), thus the necessary condition for
instability is quite likely to be satisfied.
As discussed above, Haynes et al. (1991) denoted
Ld/L as Hr/HR, because Ld/L5NHr/f0L5Hr/HR. In-
spection of (8) and (11) reveals that the necessary con-
dition for instability depends on the second and fourth
meridional derivatives of the wave forcing, hence we be-
lieve that Ld/L is a better instability measure thenHr/HR.
Figure 9 sketches out where the necessary condition
for instability in (11) is satisfied as a function of the wave
driving amplitude and length, using Qy 5 0.1b, b, and
2b, where tr 5 40 days, N5ffiffiffi5
p3 1022 s21, H 5 29 km,
and the forcing is centered at f0 5 508 such that Ld ’138. Clearly, Qy has a stabilizing effect; the larger it is,
the more stable the flow is. The nonlinear shaded gray
indicates the region where the Rossby number exceeds 1,
hence the QG approximation breaks down and the QG
downward control limit is no longer applicable. Note,
however, that the flow is likely to go unstable before it
reaches this limit. The boldface cross in the figure denotes
the amplitude and meridional scale of the perturbation
generated by the GFDLOGW scheme, shown in Fig. 3a.
Clearly, this wave forcing had to be compensated to yield
a sensible mean state.
b. Verification of the theory in the model
We next test the hypothesis that compensation is re-
lated to instability in our AGCM, where we can explore
FIG. 9. The necessary condition for instability, as in (11), as
a function of the wave driving’s amplitude A, meridional extent
L, and background PV gradient Qy, where tr 5 40 days,
N5ffiffiffi5
p3 1022 s21,H5 29 km, centered atf05 508. The solid lines
correspond to instability thresholds for Qy 5 0.1b, b, and 2b. To
the lower-right, the flow is likely to be stable and to the upper
left, the flow is likely to be unstable. The shaded gray area indicates
the nonlinear region where the downward control limit is no longer
applicable. The open circles indicate the different A and L values
that we explored in our AGCM. The thick cross denotes the am-
plitude and meridional scale of the perturbation generated by the
GFDL OGW scheme, shown in Figure 3a.
DECEMBER 2013 COHEN ET AL . 3793
the sensitivity of compensation to the amplitude A and
meridional scale L of the perturbation. We consider
a specific example in which the wave driving perturba-
tion G1 is
G1(y,p)5
(2A
2f11 cos[p(y2 y0)/L]g , if jy2 y0j#L, p1# p# p2
0, otherwise.
(16)
This compactly supported wave forcing G1 is shown by
the black contours in Fig. 8, for a case with A 5 2 31026m s22, L 5 58, p1 5 0.5 hPa and p2 5 30 hPa (such
that H 5 29 km), centered around y0 5 508N. Note that
for this cosine-shaped anomaly, the total width of the
torqueG1 is 2L, but the scale on which the torque varies
(the half-width) is L. In section 3 we showed that the
compensation is sensitive to the wave forcing amplitude
and now want to verify the analytic prediction that the
compensation is sensitive to the meridional extent of the
wave forcing.
We use the default configuration of our model, where
only the NOGW is present, and consider the response to
the steady analytic wave forcingG1. We varyA andL to
keep the total torque (proportional to AL) constant,
using the valuesA5 23 1026m s22 andL5 58 to set theoverall amplitude. Then,L is varied from 2.58 to 258. Theopen circles in Fig. 9 indicate the different A and L
values that we explored, hoping to cross the boundary
from stability to instability. All integrations were pre-
formed over 20 000 days and compared against an un-
perturbed control integration of equivalent length.
Figure 7a (see labels ‘‘L5 2.5,’’ ‘‘L5 5.0,’’ . . . , ‘‘L5 25.0’’)
show that the compensation systematically decreases with
L, despite the fact that the total torque is held constant.
Consistent with the scaling theory, the torque is largely
compensated until the width exceeds approximately 158.
We were somewhat surprised that the extremely wide tor-
ques (which are quite weak) were still partially compen-
sated. This may be because the wide torques push farther
into the subtropics, where PVgradients areweak, as seen in
Fig. 8.
Figure 7b shows the bootstrap kernel density estima-
tion of the compensation for the different experiments.
It shows that the larger the meridional extent of the
kernel density is, the larger the uncertainty is. We found
that uncertainly increases substantially in the cases with
weak compensation; this necessitates the long 20000-day
integrations. For these cases, the torque was stable, but
so weak that it was practically inconsequential in com-
parison to the resolved wave driving. The uncertainty is
less in all other integrations shown in Fig. 7a, largely
because the torques are stronger in these experiments.
5. Interactions between NOGW and resolved waves
Wehave extensively explored the effect of theOGWD,
but have not yet discussed the impacts of the NOGWD.
In the following we consider the impact of changing from
the Rayleigh drag of the Polvani and Kushner (2002)
model, whichwas envisioned as a crudeNOGWDscheme,
to theAlexander andDunkerton (1999) scheme. Figure 10
illustrates the differences between integrations with the
parameterized NOGWD and corresponding integrations
FIG. 10. The residual-mean streamfunction (109 kg s21) at 70 hPa, as a function of latitude, showing the difference
between an integration with NOGWD and a similar simulation with the Rayleigh drag, with (a) g 5 4 and (b) g 56Kkm21. The labels correspond to Fig. 6.
3794 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
with theRayleigh drag, wherewe define the ‘‘gravitywave
perturbation’’ as the difference between the Rayleigh
drag and theAlexander–DunkertonNOGWD.Twocases
are considered: one in which the polar vortex is compar-
atively weak (vortex lapse rate parameter g 5 4Kkm21)
and one inwhich the vortex is stronger (g5 6Kkm21). In
the g 5 4Kkm21 case, the overall change is about 0.1 3109 kg s21 at 70hPa, or about 5%of the total circulation at
this height. Clearly, not all the NOGWD perturbation is
compensated. Indeed, the compensation measure C 50.466 0.03, suggesting that the NOGWD perturbation is
only moderately compensated. Closer inspection of Fig.
10a, however, suggests that the overall response cannot be
simply linked to change in the NOGWD. By itself, the
change in GNOGW would create a lot of small-scale
structure, which is largely canceled out by opposite-sign
resolved wave driving perturbations. The total change
exhibits a broad structure, consistent with the mean cir-
culation, and maintains a stable PV gradient.
In the g 5 6Kkm21 case, the change is larger than
before—about 15% of the total circulation. This is an
example of an almost noncompensating case:C5 0.2160.04. In the Southern Hemisphere we can clearly see
some compensation, but in the Northern Hemisphere
the situation is quite complicated. There is still significant
anticorrelation between the change in the EPFD and the
NOGWD/Rayleigh drag, suggestive of compensation on
small scales. The overall change in EPFD on the large
scales, however, is to amplify the NOGWD perturbation,
increasing the circulation. This suggests that a naive
decomposition of the downward control response un-
derestimates the importance of theNOGWD.TheNOGW
schemeweakens the polar vortex relative to the integration
with Rayleigh friction, which in turn can increase the net
resolved wave driving of the stratosphere (Gerber 2012).
The nonlinearity of the response suggests the danger of
linearly attributing the overall BDC to different forms
of wave drivings, even when compensation is weak.
The difference in compensation between the OGWD
and NOGWD is likely due to the difference in their me-
ridional structure. The meridional scale of OGW wave
drag is tied to variations in subgrid-scale topography and
the surface winds, leading to sharp torques in the strato-
sphere that are likely to drive instability. The Alexander
and Dunkerton (1999) NOGWD is meant to capture
nonlocalized sources, convection, fronts, etc., and there-
fore generates a broader torque that is less likely to be
compensated.
6. Summary and conclusions
A new idealized modeling framework was developed
to study the interactions between resolved and
parameterized waves in the stratosphere. Orographic
and nonorographic gravity wave parameterization
schemes were implemented in a GFDL dynamical core
driven by otherwise idealized forcing. The new model
preserves the key characteristics of the troposphere–
stratosphere coupled system explored in previous ideal-
ized studies (e.g., Polvani and Kushner 2002; Gerber and
Polvani 2009) but now includes physically based mo-
mentum-conserving gravity wave parameterizations.
Based on reduced-wavenumber models, Holton (1984)
and McLandress and McFarlane (1993) suggested that
there can be substantial interaction between param-
eterized wave drag and the resolved wave driving in
the mesosphere. We extended this observation to the
stratosphere by perturbing the orographic gravity wave
scheme in our model and explore the implications of this
interaction to downward control. If the OGW drag is
increased or decreased in particular region, the resolved
wave drag (or Eliassen–Palm flux divergence) responded
to compensate for the change, so that there is little net
change in the total wave driving. Thus a decomposition of
theBDC into its wave-driven componentsmay imply large
changes in the role of parameterized and resolved waves,
but the overall circulation is remarkably robust. We found
that this compensation is fairly robust to changes in the
boundary conditions and radiative equilibrium fields, and
so is independent on the details of the parameterized or
resolved wave driving.
We interpret the compensation process as a response
of the resolved waves to maintain a ‘‘sensible,’’ stable
circulation. An analysis of downward control in the QG
limit suggests that strong and/or narrow wave forcing is
likely to drive an unstable circulation. The Rossby ra-
dius of deformation appears as a natural parameter in
the stability analysis, and perturbations on scales L
smaller than this are quite likely to go unstable, as the
QG-PV scales withL24 in this limit. In addition, we have
found that in this limit, the instability criterion is pro-
portional to the square of the wave forcing height scale;
that is, the larger the height extent becomes, the more
likely the flow will go unstable. We confirmed this in-
tuition with a systematic study using the nonlinear model.
Keeping the total wave forcing constant, we found that
compensation increases with decreasingmeridional scale.
An important question is whether compensation oc-
curs in comprehensive GCMs. The scale of gravity wave
drag perturbations is controlled by small-scale topog-
raphy and the structure of surface winds, which tend
to vary on much smaller scales relative to Ld of the
stratosphere. Hence, we expect compensation to occur.
There is indirect evidence in a recent study byMcLandress
et al. (2012) using the Canadian Middle Atmosphere
Model (CMAM). They found a remarkable degree of
DECEMBER 2013 COHEN ET AL . 3795
compensation to an OGWD perturbation located
around 508 (see Figs. 11–15 in their paper), although this
was not the focus of this study. There is also a hint of
substantial interaction in climate change scenarios of
CMAM as well (Shepherd and McLandress 2011).
Compare their Fig. 2b to Fig. 5b of this paper: the neg-
ative correlation between the OGWD and the EPFD is
the compensation signature.
The fact that compensation is more likely to occur for
wave forcings of short meridional extent led us to sus-
pect that it may be easier to find it in the Southern
Hemisphere. Here the meridional structure of small-
scale topography exhibits fine scales, particularly on the
boundaries of the Southern Ocean with Patagonia and
the Antarctic Peninsula. We obtained output from a
comprehensive atmospheric GCM, ECHAM6, the at-
mosphere component of the MPI-ESM-MR model, de-
veloped at the Max Planck Institute for Meteorology
(Stevens et al. 2012). Figure 11 shows cross sections of
the residual streamfunction at 70 hPa for each season
averaged over a 2-yr period. As expected, the winter and
spring hemispheres show larger OGW activity. Com-
pensation between the parameterized OGWD and the
resolved waves is clearly evident in the negatively cor-
related peaks between the OGWD (red) and EPFD
(blue) streamfunction, especially in the Southern Hemi-
sphere. The forcing associated with GWD in ECHAM6
was generally on the order of 1025m s22 and varies over
regions 58–108 wide, and so it is well into the nonlinear
regime suggested by Fig. 9.
It is known that the current gravity wave parameter-
izations underestimate the temporal variability (or in-
termittency) of GWdriving seen in observations (Geller
et al. 2013). Parameterizations generally smooth out this
intermittency, providing a more even torque in time. If
actual gravity wave breaking events tend to involve very
strong torques on short temporal scale, onemight expect
evenmore significant interaction with larger-scale waves
through the instability mechanism explored in this text.
The strong interaction between resolved and param-
eterized waves has implications for both the modeling
and the interpretation of the stratospheric circulation.
FIG. 11. Cross section of the residual mass meridional streamfunction at 70 hPa for the years 1990/91 using
ECHAM6 data for the different seasons. The compensation ‘‘signature’’ is highlighted by a thick black arrow.
Compare this figure to Figs. 6 and 10. As discussed in the text, the compensation signature is evident in the Southern
Hemisphere, especially during (d) austral spring. The labels correspond to Fig. 6.
3796 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 70
First, the total wave forcing in GCMs is often achieved
by tuning the GWD to obtain a reasonable zonal-mean
basic state (e.g., zonalwind, temperature)—aprocess that
is typically not well documented but known to be a chal-
lenge. Compensation between resolved and parameter-
ized waves implies that biases in the simulation of one
component (i.e., the OGWD) can be masked by biases in
other components. This makes it particularly difficult to
identify and correct model biases. Getting the tuning
right, however, is important: Sigmond and Scinocca
(2010) found that subtle differences in the OGWD can
have a significant impact on the tropospheric response to
anthropogenic forcing.
A second implication is how we should interpret the
driving of the Brewer–Dobson circulation. The BDC is
often decomposed linearly into different wave-driven
components using the downward control principle. The
strong interaction between the resolved and parame-
terized wave driving indicates that this separation may
provide an incomplete, or even misleading, illustration of
the meridional circulation physics. This may explain the
intermodel variance in the role of resolved and parame-
terized waves in driving the BDC.Models do tend to agree
more on the total strength of the circulation than of
individual components (e.g., Butchart et al. 2011, their
Fig. 10). Compensation will tend to reduce the impact of
difference inGWperturbation on the total circulation. This
may also be relevant to analysis of future changes in the
BDC, where GCMs agree on an increase in the circulation
but disagree on the role of GW versus resolved waves.
Tomove forward, we recommend that modeling groups
report more detailed information on the resolved and
parameterizedwave driving, which can be used to quantify
the degree of compensation in their simulations. Monthly
and zonal-mean EPFD and parameterized wave torques
would be sufficient, allowing for the calculations shown in
Fig. 11. There is also a need for observations (e.g., detailed
measurements of temperature variance as a function of
height) to better constrain the structure of GWD.
Even in cases with weak compensation as discussed in
section 5, we still find a very nonlinear response of the
resolved wave driving to changes in GWD, demonstrat-
ing that downward control is not additive. The idea that
downward control is not linear in this sense puts some
limitations on the controllability of a given forcing. In that
context, downward control is limited by admissible
stratospheric wave forcing, yet to be defined properly.
Acknowledgments.We thank Felix Bunzel and Hauke
Schmidt at the Max Planck Institute for Meteorology for
providing the ECHAM6 output. This research was sup-
ported in part by Grant AGS-0938325 from the National
Science Foundation to New York University. We also
thank Olivier M. Pauluis, Tiffany A. Shaw, M. Joan
Alexander, and two anonymous reviewers for comments
and suggestions.
APPENDIX
Gravity Wave Scheme Modifications
We specify the net momentum flux at the source level
as a stress (Pa), as suggested byAlexander andDunkerton
(1999) and Donner et al. (2011). To the best of our
knowledge, however, the net momentum flux at the
source level in AM3’s OGW parameterization appears to
be interpreted as a flux (m2 s22) with a tuning factor in
order to get the correct Northern Hemisphere midlatitude
stress. The AM3 settings were corrected by taking into
account the density at the source level. This change does
not have a significant impact on our integrations.
To avoid nonconservative wave driving, which can
affect the stratosphere and troposphere downward
control (Shepherd and Shaw 2004; Shaw and Shepherd
2007), we have added a simple condition to smoothly
deposit all residual OGWD within the five layers above
0.5 hPa. Depositing it all in the uppermost level, as sug-
gested by Shaw et al. (2009), led to instability, possibly
because our model’s top is substantially higher than that
in their study. In the configuration of the scheme used in
AM3, however, gravity wave flux above 30hPa was al-
lowed to escape to space. We also modified the NOGW
parameterization to ensure that any residual flux was
deposited in the uppermost level, although this had little
impact on the integrations owing to the model’s high top.
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